(1) Since \(\omega _1, \omega _2\)  are locally normal, we can define a normal state \([\omega _1\times \omega _2]_n\)  on \(W^*(G_n\times G_n) =W^*(G_n)\,\bar{\otimes }\,W^*(G_n)\)  to be \((\omega _1\!\upharpoonright _{W^*(G_n)})\,\bar{\otimes }\,(\omega _2\!\upharpoonright _{W^*(G_n)})\) . By definition, it is clear that \([\omega _1\times \omega _2]_{n+1}\)  agrees with \([\omega _1\times \omega _2]_n\)  on \(W^*(G_n\times G_n)\) . Thus, the inductive limit \(\omega _1\times \omega _2 := \varinjlim [\omega _1\times \omega _2]_n\)  defines the desired locally normal state on \(\mathcal {B}(G\times G)\) , which clearly agrees with \(\omega _1\otimes \omega _2\)  on \(\mathcal {B}_\infty (G)\otimes \mathcal {B}_\infty (G)\) . The uniqueness of \(\omega _1\times \omega _2\)  follows from its local normality.
(2) Since \(\hat{\Delta } = \varinjlim \hat{\Delta }_n\)  is clearly locally normal and (REF ), one can easily confirm that \(\chi _1\cdot \chi _2\)  falls into \(K_{-1}^\mathrm {ln}(\vartheta ^t)\) .
Let \(n \ge 0\)  and \(z \in \mathfrak {Z}_n\)  be arbitrarily chosen and fixed. By (REF ) we have
\((\hat{\Delta }_n\otimes \mathrm {id})(u_z)=(\Phi _{U_\bullet }^{(2)}{}^{-1}\otimes \mathrm {id})\Big (\bigoplus _{(z_1,z_2)}((U_{z_1}\times U_{z_2})(P^{(z_1,z_2)}_z \otimes 1))\Big ),\) 
where \((z_1,z_2)\)  runs as long as \(U_z\)  appears in \(U_{z_1}\times U_{z_2}\)  as dirrect summands and \(P_z^{(z_1,z_2)}\)  is the orthogonal projection onto the whole invariant subspace corresponding to \(U_z\) . By the normality of \([\chi _1\times \chi _2]_n\)  we observe that
\(&(\chi _1\cdot \chi _2)(u_z)=([\chi _1\times \chi _2]_n\circ \hat{\Delta }_n\otimes \mathrm {id})(u_z) \\&=\sum _{(z_1,z_2)}([\chi _1\times \chi _2]_n\otimes \mathrm {id})\big ((\Phi _{U_\bullet }^{(2)}{}^{-1}\otimes \mathrm {id})((U_{z_1}\times U_{z_2}) (P^{(z_1,z_2)}_z \otimes 1))\big ) \\&=\sum _{(z_1,z_2)} \chi _1(z_1)\,\chi _2(z_2)\frac{1}{\mathrm {Tr}(\pi _{U_{z_1}\times U_{z_2}}(\rho _n))}(\mathrm {Tr}\otimes \mathrm {id})((\pi _{U_{z_1}\times U_{z_2}}(\rho _n)\otimes 1)((U_{z_1}\times U_{z_2})(P^{(z_1,z_2)}_z \otimes 1))\) 

and the (operator) norm of each term of the sum in the last expression is not greater than
\(\chi _1(z_1)\,\chi _2(z_1)\,\frac{\mathrm {Tr}(\pi _{U_{z_1}\times U_{z_2}}(\rho _n)P_z^{(z_1,z_2)})}{\mathrm {Tr}(\pi _{U_{z_1}\times U_{z_2}}(\rho _n))}.\) 
Consequently, we obtain that
\(&(\chi _1\cdot \chi _2)(u_n)=\sum _{z\in \mathfrak {Z}_n} (\chi _1\cdot \chi _2)(u_z) \\&=\sum _{(z_1,z_2)} \sum _z \chi _1(z_1)\,\chi _2(z_2)\frac{1}{\mathrm {Tr}(\pi _{U_{z_1}\times U_{z_2}}(\rho _n))}(\mathrm {Tr}\otimes \mathrm {id})((\pi _{U_{z_1}\times U_{z_2}}(\rho _n)\otimes 1)(U_{z_1}\times U_{z_2})(P^{(z_1,z_2)}_z \otimes 1)) \\&=\sum _{(z_1,z_2)} \chi _1(z_1)\,\chi _2(z_2)\frac{1}{\mathrm {Tr}(\pi _{U_{z_1}\times U_{z_2}}(\rho _n))}(\mathrm {Tr}\otimes \mathrm {id})((\pi _{U_{z_1}\times U_{z_2}}(\rho _n)\otimes 1)(U_{z_1}\times U_{z_2}))\) 

in norm, and the last expression is easily shown to be \(\chi _1(u_n)\,\chi _2(u_n)\)  since
\(U_{z_1}\times U_{z_2} = (U_{z_1})_{13}(U_{z_2})_{23}, \qquad \pi _{U_{z_1}\times U_{z_2}}(\rho _n) = \pi _{U_{z_1}}(\rho _n)\otimes \pi _{U_{z_2}}(\rho _n)\) 
(see [1]}).
It is easy to see that \(\hat{\varepsilon }\)  falls into \(K_{-1}^\mathrm {ln}(\vartheta ^t)\)  and \(\hat{\varepsilon }\cdot \chi = \chi = \chi \cdot \hat{\varepsilon }\)  holds for every \(\chi \in K_{-1}^\mathrm {ln}(\vartheta ^t)\) .
(3) By \(\hat{R} = \varinjlim \hat{R}_n\)  and (REF ), one easily sees that \(\chi ^*\)  falls into \(K_{+1}^\mathrm {ln}(\vartheta ^t)\) .
Thanks to [1]} we observe that
\(U_z^* = (j\otimes \mathrm {id})(U_z^c)= \pi _{U_z}(\rho _n)^{-1/2}(j\otimes \mathrm {id})(\bar{U}_z)\pi _{U_z}(\rho _n)^{1/2},\) 

and hence
\(\chi (u_z)^*&=\chi (z)\,\tau _z(u_z) \\&=\chi (z)\,\frac{1}{\mathrm {Tr}(\pi _{U_z}(\rho _n))} (\mathrm {Tr}\otimes \mathrm {id})\big ((\pi _{U_z}(\rho _n)\otimes 1)(j\otimes \mathrm {id})(\bar{U}_z)\big ) \\&=\chi (z)\,\frac{1}{\mathrm {Tr}(\pi _{\bar{U}_z}(\hat{R}(\rho _n)))} (\mathrm {Tr}\otimes \mathrm {id})\big ((\pi _{\bar{U}_z}(\hat{R}(\rho _n))\otimes 1)(\bar{U}_z)\big ) \\&=\chi ^*(\bar{z})\,\frac{1}{\mathrm {Tr}(\pi _{\bar{U}_z}(\rho _n^{-1}))} (\mathrm {Tr}\otimes \mathrm {id})\big ((\pi _{\bar{U}_z}(\rho _n^{-1})\otimes 1)(\bar{U}_z)\big ) \\&=\chi ^*(\bar{z})\,\frac{1}{\mathrm {Tr}(\pi _{U_{\bar{z}}}(\rho _n^{-1}))} (\mathrm {Tr}\otimes \mathrm {id})\big ((\pi _{U_{\bar{z}}}(\rho _n^{-1})\otimes 1)(U_{\bar{z}})\big )\) 

by [1]} and by (REF ) and (REF ). The uniqueness of normal \((\vartheta _{\bar{z}}^t,+1)\) -KMS state on \(\bar{z}W^*(G_n) \cong B(\mathcal {H}_{\bar{z}})\)  implies that \(\chi (u_z)^* = \chi ^*(u_{\bar{z}})\) . Thus we conclude that
\(\chi (u_n)^* = \sum _{z\in \mathfrak {Z}_n} \chi (u_z)^* = \sum _{z\in \mathfrak {Z}_n} \chi ^*(u_{\bar{z}}) = \chi ^*(u_n).\) 
The other assertion is trivial now.
